Computer Science/Discrete Mathematics Seminar II

Miller computed the character table of $S_n$ for some fairly large n's and noticed that almost all of the entries were even. Based on this, he conjectured that the proportion of even entries in the character table of $S_n$ tends to $1$ as $n\to\infty$. I'll present two proof of this conjecture, one of which shows more generally that almost every entry of the character table of $S_n$ is divisible by $p$ for all primes $p$ up to $13$. Interestingly, this proof naturally breaks down past $p=13$. I will also discuss the problem of determining the density of zeros in the character table of $S_n$. Miller also has computational data suggesting that this density is a positive number less than $1$. No background on the representation theory of $S_n$ will be assumed--since the talk is two hours, I will spend the first part introducing this topic.